Problem: Simplify the following expression and state the condition under which the simplification is valid. $t = \dfrac{z^2 - 81}{z - 9}$
Explanation: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = z$ $ b = \sqrt{81} = -9$ So we can rewrite the expression as: $t = \dfrac{({z} {-9})({z} + {9})} {z - 9} $ We can divide the numerator and denominator by $(z - 9)$ on condition that $z \neq 9$ Therefore $t = z + 9; z \neq 9$